Theorem reurmo 2772

Description: Restricted existential uniqueness implies restricted "at most one." (Contributed by NM, 16-Jun-2017.)

Assertion

Ref	Expression
reurmo	⊢ (∃!𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑)

Proof of Theorem reurmo

Step	Hyp	Ref	Expression
1		reu5 2770	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐴 𝜑))
2	1	simprbi 275	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4  ∃wrex 2529  ∃!wreu 2530  ∃*wrmo 2531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-rex 2534  df-reu 2535  df-rmo 2536

This theorem is referenced by: (None)